Quadrature-phase shift keying (QPSK) is a modulation scheme that encodes data into one of four phase values for a carrier signal. Since there are four possible phase values, two data bits can be encoded into each modulated signal. Typically the four phase values are ninety degrees apart from each other, e.g., 0°, 90°, 180°, and 270°, or 45°, 135°, 225°, and 315°.
Conventional narrowband QPSK encoding is accomplished by using two separate carrier signals, which are used to encode an in-phase (I) path and a quadrature-phase (Q) path. The I and Q phase paths each have a separate stream of data encoded into them, and the resulting two data paths are added together to form a final QPSK signal. This final QPSK signal will have one of the four possible phase values, indicating the four possible bit value combinations, i.e., 00, 01, 10, or 11. This could be done using gray coding, natural coding, or any desired type of QPSK coding.
To function properly, the in-phase and quadrature-phase signals must be orthogonal to each other and must each have the same signal energy. If a function ƒI(t) defines a signal used for an I path, and a function ƒQ(t) defines a signal used for a Q path, then the requirement of orthogonality can be shown by Equation (1):
                                          ∫                          -              ∞                        ∞                    ⁢                                                    f                I                            ⁡                              (                t                )                                      ·                                          f                Q                            ⁡                              (                t                )                                      ·                                                  ⁢                          ⅆ              t                                      =        0                            (        1        )            
In other words, if the two signals are multiplied by each other and integrated over time, the result should be zero.
The requirement of equal energy can be shown by Equation (2):
                                          ∫                          -              ∞                        ∞                    ⁢                                                    f                I                2                            ⁡                              (                t                )                                      ·                                                  ⁢                          ⅆ              t                                      =                              ∫                          -              ∞                        ∞                    ⁢                                                    f                Q                2                            ⁡                              (                t                )                                      ·                                                  ⁢                          ⅆ              t                                                          (        2        )            
In other words, if the two signals are each multiplied by themselves and integrated over time, the results should be equal. This means that regardless of whether you're encoding digital 1's or 0's into the two waveforms, they'll have the same energy and therefore the same signal-to-noise ratio at the receiver.
In a narrowband system, sine and cosine signals are typically used for ƒI(t) and ƒQ(t). And though it's hard to make these two signals perfectly orthogonal for any given transmission, they make a close approximation. A large number of cycles are used, and the signals are tapered off at the end, which makes the resulting waveforms close enough to orthogonal and close enough to having the same signal energy that the requirements of Equations (1) and (2) are met to within acceptable tolerances.
However higher bandwidth systems (e.g., wideband and ultrawide bandwidth systems) use very short duration signals (which may also be called symbols or waveforms) to pass a pair of data bits. The duration of these symbols is so short that sine and cosine signals will not fulfill the requirements of orthogonality and equal energy. Therefore, it is desirable to provide short-duration signals that are appropriate for encoding high-speed QPSK data.